> R Hansen says: > >Then prove it. Surely you accept that everyone on > this list understands to some degree and acknowledges > abstract analytical (non visual) thinking. I think > the task is up to you to prove that this mysterious > non abstract visual thinking exists and is up to the > task of mathematics. I have found no evidence of it > > Your position is utterly bizarre. There are books and > papers written about visualization in mathematics. I > don't care how much you want to torture the meaning > of the word with your idiosyncratic talk. > > Cheers, > Joe N
I think Robert falls into my first category.
I also think he is confusing thinking with communicating, and mental imagery with visual aids. Mental imagery and other forms of thinking are fluid, dynamic, abstract processes. Communicating those processes, whether through mathematical symbols or visual aids, always falls short, reducing our rich inner life to something relatively more concrete, static, and 2-dimensional.